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Chapter 12: Problem 40

Show that the Second Derivative Test is inconclusive when applied to the following functions at(0,0) Describe the behavior of the function at the critical point. \(f(x, y)=x^{2} y-3\)

Short Answer

Expert verified
Question: Show that the Second Derivative Test is inconclusive when applied to the function \(f(x, y) = x^2y - 3\) at the point \((0,0)\), and describe the behavior of the function at the critical point. Answer: The Second Derivative Test is inconclusive for the function \(f(x, y) = x^2y - 3\) at the point \((0, 0)\) because the determinant of the Hessian matrix at this point is equal to 0. However, by examining the function and its second-order derivatives, we can observe that the function exhibits different behavior depending on the direction in which we move away from the origin. This suggests that the point is likely a saddle point.

Step by step solution

01

Find the critical points of f(x, y)

To find the critical points, we must compute the first-order partial derivatives of the function with respect to x and y: $$ \frac{\partial f}{\partial x} = 2xy \\ \frac{\partial f}{\partial y} = x^2 $$ Now, set these partial derivatives equal to zero and solve for x and y: $$ 2xy = 0 \\ x^2 = 0 $$ The only solution to this system of equations is \((x, y) = (0, 0)\). Therefore, the critical point of the function is at \((0, 0)\).
02

Apply the Second Derivative Test

To apply the Second Derivative Test, we must compute the second-order partial derivatives of the function and form the Hessian matrix: $$ \frac{\partial^2 f}{\partial x^2} = 2y \\ \frac{\partial^2 f}{\partial y^2} = 0 \\ \frac{\partial^2 f}{\partial x \partial y} = 2x $$ The Hessian matrix H is: H = $\begin{bmatrix} 2y & 2x \\ 2x & 0 \end{bmatrix} $ Now, evaluate the Hessian matrix at the critical point \((0, 0)\): H(0, 0) = $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} $ The determinant of H(0, 0) is equal to \((0)(0) - (0)(0) = 0\). Since the determinant is zero, the Second Derivative Test is inconclusive.
03

Describe the behavior of the function at the critical point

Although the Second Derivative Test is inconclusive, we can still make observations about the function and its critical point "\((0, 0)\)" by examining the function and its second-order partial derivatives. We can see that the second-order partial derivatives are linear functions of x and y. This implies that as we move away from the origin, the change in the concavity (and thus the function's behavior) varies depending on the direction in which we move. For instance, moving along the x-axis would cause the mixed partial derivative \(f_{xy}\) to remain zero, while the second-order derivative \(f_{xx}\) would change linearly with x. Thus, the function exhibits different behavior along different directions around the critical point \((0, 0)\). This also implies that the point is likely a saddle point, as the inconclusive Second Derivative Test and the linear dependence of the second-order partial derivatives suggest. To summarize, we have shown that the Second Derivative Test is inconclusive when applied to the function \(f(x, y) = x^2y - 3\) at the point \((0, 0)\) and that the function exhibits different behavior along different directions around this critical point.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Points
Critical points in calculus represent the locations on a graph where a function has either a local maximum, local minimum, or a saddle point. To locate critical points, one must first determine the first-order partial derivatives of the function and set them equal to zero.

For the given function \(f(x, y) = x^2y - 3\), the critical points are the solutions to \(2xy = 0\) and \(x^2 = 0\). The only solution to this system is \((x, y) = (0, 0)\), making it a critical point of the function. Understanding where these points occur and their nature is essential in multi-variable calculus as they are often related to the extremes of the function.

However, finding a critical point does not tell us whether it's a maximum, minimum, or saddle point. To determine this, one would usually use the Second Derivative Test, which in this particular case is inconclusive. We then need to rely on other methods, such as examining the behavior of the function around the critical point for further information.
Partial Derivatives
Partial derivatives represent the rate of change of multivariable functions with respect to one variable while keeping the other variables constant. They are fundamental in identifying the slope of the function in the direction of that variable.

For the function \(f(x, y) = x^2y - 3\), the first-order partial derivatives calculated are \(\frac{\partial f}{\partial x} = 2xy\) and \(\frac{\partial f}{\partial y} = x^2\). These derivatives show how \(f(x, y)\) changes as either \(x\) or \(y\) varies, while the other remains fixed.

Understanding these rates of change is crucial in predicting the behavior of the function. When we set the first-order partial derivatives equal to zero to find where these rates change sign, we identify the possible critical points, as we've done in our exercise. Appreciating the notion of varying independently in each direction is key to grasping how multivariable functions behave in higher dimensions.
Hessian Matrix
The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function. It is a pivotal tool in multi-variable calculus, especially for analyzing the curvature and the local extrema of functions.

Applied to our function \(f(x, y)\), the Hessian matrix crucially informs us about concavity and convexity near the critical point. Here, the Hessian is composed of the second-order partial derivatives \(\frac{\partial^2 f}{\partial x^2} = 2y\), \(\frac{\partial^2 f}{\partial y^2} = 0\), and \(\frac{\partial^2 f}{\partial x \partial y} = 2x\).

Evaluating the Hessian matrix at the critical point \((0, 0)\) gives us a matrix of zeros, leading to a zero determinant. This zero determinant means the Second Derivative Test fails to provide conclusive information regarding the nature of the critical point. In such cases, the Hessian matrix tells us that the function's curvature is sensitive to the direction we choose to analyze, hinting towards more complex behavior like a saddle point, which requires further exploration beyond the scope of the Second Derivative Test.

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